light scattering by gaussian particles: rayleigh-ellipsoid...

Journal of Quantitative Spectroscopy &Radiative Transfer 63 (1999) 277}303

Light scattering by Gaussian particles: Rayleigh-ellipsoidapproximation

A. Battaglia!,*, K. Muinonen", T. Nousiainen#, J.I. Peltoniemi$

!University of Ferrara, via Paradiso, 12, I-44100 Ferrara, Italy"Observatory, University of Helsinki, P.O. Box 14, FIN-00014 U. Helsinki, Finland

#Department of Meteorology, University of Helsinki, P.O. Box 4, FIN-00014, U. Helsinki, Finland$Finnish Geodetic Institute, P.O. Box 15, FIN-02431 Masala, Finland

Abstract

We study absorption and scattering by irregularly shaped Gaussian random particles in theRayleigh-ellipsoid approximation. For a given sample shape, we determine the best-"tting ellipsoid asthe equal-volume ellipsoid with the largest volume overlapping the sample shape. We present an e$cientmethod for calculating such ellipsoids for Gaussian particles and characterize the goodness of the approxi-mation with the complementary volume. We study the scattering properties of Gaussian particles muchsmaller than the wavelength with di!erent complex refractive indices, comparing the Rayleigh-ellipsoidapproximation to the Rayleigh-volume, discrete-dipole, and second-order perturbation approximations,and to the computations using the variational volume integral equation method. Our new method canprove valuable in microwave remote sensing of terrestrial ice clouds: crystalline structures are oftenelongated with dimensions in the Rayleigh domain for typical radar frequencies. ( 1999 Elsevier ScienceLtd. All rights reserved.

1. Introduction

Scattering of light by Gaussian particles has recently been studied, e.g., in the ray opticsapproximation [1,2] and in the Rayleigh and Rayleigh}Gans regimes [3]. In what follows, the

*Corresponding author. Tel.: #39-51-639-9569; fax: #39-51-639-9658E-mail address: batta@hail."sbat.bo.cnr.it (A. Battaglia)

0022-4073/99/$ - see front matter ( 1999 Elsevier Science Ltd. All rights reserved.PII: S 0 0 2 2 - 4 0 7 3 ( 9 9 ) 0 0 0 2 0 - 5

Rayleigh-ellipsoid approximation [4,5] (REA) is established for Gaussian particles, and parti-cularly compared to the so-called Rayleigh-volume approximation (RVA).

In RVA [3], the ensemble-averaged absorption and scattering cross sections are proportional tothe average volume and volume squared, respectively, while the scattering matrix is that ofa Rayleigh sphere. RVA follows from the Rayleigh}Gans approximation when the particle size ismuch smaller than the wavelength of incident light. Thus, in RVA, ensembles of particles withsimilar volume distributions produce similar absorption and scattering characteristics. It is worthnoting that, in RVA, similar ensemble-averaged scattering and absorption cross sections onlyrequire similar "rst and second moments of volume.

In RVA, the composition and the shape of the particle do not a!ect the scatteringmatrix, whereas they do a!ect the absolute value of the scattering cross section. Such an ap-proximation can be expected to work except for high refractive indices or very elongatedparticles.

Our goal, in the Rayleigh-ellipsoid approximation, is to see whether, approximating the shapesof small particles with ellipsoids, we can obtain a better approximation of the scattering properties.In particular, since ellipsoids can be treated in the resonance region [with the ¹-matrix method (seeRef. [6]), or, but only for spheroids, the separation of variables technique (see Ref. [7]), forexample], it is one of our goals to establish a mathematically sound de"nition for the best-"ttingellipsoid.

We note that the full electrostatics approximation for Gaussian particles would requirethe detailed solution of the Laplace equation with the necessary boundary conditions.For ellipsoids, the Rayleigh-ellipsoid approximation is equivalent to the electrostatics ap-proximation.

The interest towards Gaussian shapes has increased steadily and, accordingly, they havebeen studied extensively in the last few years; they actually represent one of the most powerfulattempts to describe the size and shape distribution of small particles, following a statistical pointof view.

Up till now Gaussian shapes have been used to model cosmic dust particles (e.g. [1,8]),oscillation of raindrops [9,10] and the shapes of asteroids [11}13].

Schi!er [14] considered Gaussian particles with surface #uctuations small as compared to themean radius. In the following work, a brief summary is given on the principal mathematical aspectsof the Gaussian particles, with particular attention to the shapes used in the following simulations;for more details, see other works [1,12,15].

The second-order perturbation approximation ([14], see also Muinonen [15]) provides a fastmethod for computing ensemble-averaged scattering characteristics for Gaussian particles withsurface #uctuations small compared with the radius and the wavelength of incident light. Thecurrent work allows us to assess the validity of the perturbational approach. In addition, asa continuation of our recent work [16], we include results from the discrete-dipole approximation[17] and the variational volume integral equation technique [18].

In Section 2, we summarize the basic features of Gaussian particles and, in Section 3, theRayleigh-ellipsoid approximation. Section 4 provides background for the Rayleigh-volume, dis-crete-dipole, and second-order perturbation approximations, and the variational volume integralequation technique. Numerical methods follow in Section 5. We discuss the main results in Section6, and conclude in Section 7.

278 A. Battaglia et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 63 (1999) 277}303

2. Gaussian particle

The three-dimensional Gaussian random sphere r"r(0,u) is described by the spherical har-monics series for the so-called logradius s"s(0, u),

r(0,u)"r6 expCs(0, u)!12b2D,

s(0,u)"=+l/0

l+

m/~l

slm>

lm(0, u), (1)

where r6 and b are the mean radius and the standard deviation of the logradius, and >lm's are the

orthonormal spherical harmonics. The logradius is real-valued so that

sl,~m

"(!1)msHlm

, l"0, 1,2,R, m"!l,2,!1, 0, 1,2, l, (2)

implying Im(sl0),0.

The correlation functions of the radius and logradius Crand C

s, respectively, and the corre-

sponding variances p2 and b2 are interrelated through

p2Cr"exp(b2C

s)!1,

p2"exp(b2)!1. (3)

Note that the relative standard deviation of radius p"Jexp(b2)!1 depends only on b.The real and imaginary parts of the spherical harmonics coe$cients s

lm(m50) are independent

Gaussian random variables with zero means and variances

Var(Re(slm

))"(1#dm0

)2p

2l#1C

l,

Var(Im(slm

))"(1!dm0

)2p

2l#1C

l,

l"0, 1,2,R, m"0, 1,2, l. (4)

The coe$cients Cl50 (l"0,2,R) are the Legendre coe$cients of the logradius covariance

function &s,

&s(c)"b2C

s(c)"

=+l/0

ClP

l(cos c),

=+l/0

Cl"b2, (5)

A. Battaglia et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 63 (1999) 277}303 279

where c is the angular distance between two directions and where Csis the logradius correlation

function,

Cs(c)"

=+l/0

clP

l(cos c),

=+l/0

cl"1. (6)

For each degree l, the coe$cient Clprovides the spectral weight of the corresponding spherical

harmonics components in the Gaussian sphere. On one hand, weighting the spectrum towardhigher-degree harmonics will result in shapes with larger numbers of hills and valleys per solidangle. On the other hand, increasing the variance of the logarithmic radius will enhance the hillsand valleys radially. In practice, the series representations in Eqs. (1) and (5}6) need to be truncatedat a certain degree l

.!9su$ciently high to maintain good precision in the generation of sample

spheres.The two perpendicular slopes

s0"r0

r,

1sin0

sr"rr

r sin0, (7)

are independent Gaussian random variables with zero means and standard deviations

o"J!&(2)s

(0), (8)

where &(2)s

is the second derivative of the covariance function with respect to c. The correlationlength l

#and correlation angle !

#are de"ned by

l#"

1

J!C(2)s

(0)"

bo

,

!#"2 arcsinA

12l#B. (9)

For the present work, we have chosen two kinds of correlation functions Cs; correspondingly we

have Classes-I and II Gaussian particles. Some realizations of these two types of Gaussian particlesare shown in Fig. 1.

Class I. Particles with a modi"ed Gaussian correlation function:

Cs(c)"expA!

2l2#

sin212

cB (10)

with coe$cients for Legendre polynomials (all non-negative) as

cl"(2l#1) expA!

1l2#BilA

1l2#B, l"0,2,R, (11)


Fig. 1. Realizations of Gaussian particles with Legendre polynomial correlation functions (a) P1, (b) P

2, (c) P

3, and

with the correlation angle (d) !#"103, (e) !

#"303, and (f ) !

#"903 in a modi"ed Gaussian correlation function. The

relative standard deviation of radius is p"0.3 in all cases.


where ilis a modi"ed spherical Bessel function. Fixing the correlation function, the stochastic shape

is parameterized by p and !#. Note that decreasing !

#means lower l

#values and, accordingly,

longer spherical harmonics expansions. This implies longer computation times and more pro-nounced #uctuations on the particle's surface; generally speaking decreasing p and increasing!#lead toward the spherical shape. We will refer to these particles as Class-I particles and, for our

computations, we use !#"10, 30, 903 and p"0.1, 0.2, 0.3.

Class II. Harmonic Gaussian particles:

Cs(c)"P

k(cos c), k"1,2,3, (12)

where Pk

is the Legendre polynomial of degree k. We use particles P1, P

2and P

3with the same

values of p as for Class-I particles. In particular, the P2-particles, associated with the correlation

function Cs(c)"(3cos2 c!1)/2, generate elongated particles with elongation increasing with

increasing p. We consider this kind of Gaussian particles very interesting as they can closelyrepresent many hydrometeors.

3. Rayleigh-ellipsoid approximation

It is well known that the Rayleigh approximation for spheres is equivalent to the electrostaticsapproximation [5]: solving the electrodynamic problem is equal to solving the electrostaticproblem, because the internal and applied "elds are in phase. The dipole radiation pattern can thenbe computed from the dipole moment that is proportional to the incident "eld; the tensor ofproportionality is generally known as the polarizability.

3.1. Theory

The electrostatic problem of a dielectric sphere of radius a is well described by Jackson [19]; thepolarizability is a scalar:

a"4p a3m2!1m2#2

"4p a3e1!e

me1#2e

m

, (13)

where e1

and em

are the permittivities of the scatterer and the surrounding medium. In the sphericalpolar coordinates, ellipsoids with semiaxes a, b, c are speci"ed by

r(h,/)"abc

Jb2c2sin2h cos2/#a2c2 sin2h sin2/#a2b2 cos2h. (14)

The electrostatics problem is more complicated for ellipsoids, since the polarizability remainsa non-trivial tensor. However, in the principal axes reference frame K, the tensor is diagonal:

a"4p abcAe1~em

3em`3L1(e1~em) 0 0

0 e1~em3em`3L2(e1~em) 0

0 0 e1~em3em`3L3(e1~em)B, (15)


where the purely geometric factors ¸i(¸

1#¸

2#¸

3"1) can be computed numerically from

one-dimensional integrals,

G¸1"abc

2:=0

dq(a2#q) f (q)

,

¸2"abc

2:=0

dq(b2#q) f (q)

,

¸3"abc

2:=0

dq(c2#q) f (q)

,

f (q)"J(a2#q)(b2#q)(c2#q), (16)

Fig. 2 provides scatter plots of ¸1

and ¸2

for some Gaussian shapes.To compute the polarizability tensor in any basis we use the transformation properties of

a tensor: passing from the principal axes reference frame K to another frame K@ with an orthogonaltransformation A (x"Ax@), we will have

a@"ATaA. (17)

In the following case, the orthogonal transformation will be implemented by a rotation character-ized by the three Euler angles [20] (a

E,b

E, c

E).

From the tensor of polarizability it is possible to compute the induced dipole moment p from

p"emaE

0(18)

and then, using the known dipole radiation pattern, to "nd all the scattering characteristics forsingle particles in a given orientation. For example, for a sphere, the resulting cross sections are

C!"4

"k Im(a)"3k< ImAe1!e

me1#2e

mB, (19)

C4#!

"

k4

6pDaD2"

32p

k4<2Ke1!e

me1#2e

mK2, (20)

where < is the volume of the sphere, k"2p/j, and the scattering matrix

S"38p

k2C4#!A

12(1#cos2h

s) !1

2sin2h

s0 0

!12sin2h

s12(1#cos2h

s) 0 0

0 0 cos hs

0

0 0 0 cos hsB. (21)

Averaging over random orientations it is possible to write for ellipsoids the analogous of Eqs.(19)}(20):

CM!"4

"

k3

Im(a1#a

2#a

3), (22)

CM4#!

"

k4

18p(Da

1D2#Da

2D2#Da

3D2), (23)


Fig. 2. (a) Contour plot of the scattering cross section for an ellipsoid with the refractive index m"1.33#i0.01 in theelectrostatics approximation divided by the one computed in the Rayleigh-volume approximation as a function of ¸

1(the

smallest geometrical factor) and ¸2

(the largest factor), including a scatter plot of the best-"t ellipsoids for P2

particleswith p"0.3. Thanks to Eq. (42) the same contour plot is valid for absorption cross section. (b) As in (a) for the refractiveindex m"3#i4 and !

#"903 particles with p"0.3. Note that, since +

i¸i"1, ¸

1and ¸

2are constrained by the

conditions ¸1#2¸

251 (since ¸

2is the largest) and 2¸

1#¸

241 (since ¸

1is the smallest); therefore, only the region on

the left side of the dashed lines is allowed. It is evident that the minimum is realized at ¸1"¸

2"1/3 and that the plot is

symmetric with respect to ¸1"¸

2.


where the bar stands for the orientational average. For the Mueller matrix, we obtain

SM "ASM11

SM12

0 0

SM12

SM22

0 0

0 0 SM33

0

0 0 0 SM44B (24)

with

SM11

"

3k2CM4#!

8p12A

6!M5

#

2#3M5

cos2hsB

SM12

"

3k2CM4#!

8p12(cos2 h

s!1)

2#3M5

SM22

"

3k2CM4#!

8p12(cos2 h

s#1)

2#3M5

SM33

"

3k2CM4#!

8p2#3M

5cos h

s

SM44

"

3k2CM4#!

8pM cos h

s, (25)

where M (!124M41) is a function of the refractive index and of the axes of the ellipsoid:

M"

Re(aH1a2#aH

1a3#aH

2a3)

Da1D2#Da

2D2#Da

3D2

. (26)

Thus, the refractive index, besides entering the cross sections, also enter the scattering matrix,modifying the RVA one in Eq. (21). Particular features of the scattering matrix in Eq. (24) are thefollowing:

f the degree of linear polarization for incident unpolarized light

P"!

SM12

SM11

"

2#3M6!M

(27)

is always non-negative and has its maximum at hs"903. In general, the maximum is less than

unity;f the ratio S

22/S

11has its minimum equal to P(903) at h

s"903, and its maxima (2#3M)/(4#M)

in the forward and backward directions so there is, in general, partial depolarization of lightlinearly polarized parallel or perpendicular to the scattering plane;

f the ratio S33

/S11

has its maximum at hs"03 and its minimum at h

s"1803 equal to

$(2#3M)/(4!M), respectively, and becomes zero at hs"903. S

33(03)"S

22(03) while

S33

(1803)"!S22

(1803);f the ratio S

44/S

11behaves like the ratio S

33/S

11, but the maximum and minimum values are

$2M/(4!M);


f the linear backscattering depolarization ratio, de"ned as the ratio of the power backscattered atvertical polarization to the power backscattered at horizontal polarization for a horizontallypolarised incident "eld, has the value (1!M)/(3#2M);

f the circular backscattering depolarization ratio, de"ned as the ratio of the power backscatteredat left-hand circular polarization to the power backscattered at right-hand polarization fora left-hand circularly polarized incident "eld, has the value 2(1!M)/(2#3M).

So we can expect from the scattering matrix (24) only slight changes from the Rayleigh scatteringmatrix (21): for this reason, we have focused our attention on studying the behavior of the matrixelements only at h

s"0, 90, and 1803, which completely characterize the elements. Note that the

qualitative patterns (positions of the maxima, of the minima and of the zeros) will be conserved alsofor an average over di!erent ellipsoids.

3.2. Best-xtting ellipsoid

To study the scattering properties of Gaussian particles in the Rayleigh-ellipsoid approximation,we "rst have to de"ne the best-"tting ellipsoid for a sample shape. There are a lot of possiblemetrics and distances between two di!erent shapes. Karhunen}Loeve (see Ref. [21]) developeda statistical method for approximating a surface with an ellipsoid by taking many points N

P(thousands) on the surface of the particle and, from these, computing a covariance matrix, theeigenvectors and eigenvalues of which are connected, respectively, to the directions and the lengthsof the semiaxes. A similar approach has also been used in astrophysics to specify the orientation ofsuperclusters of galaxies. [22]. In another work [23], to model cirrus crystals, spheroidal particleswith equal volume and aspect ratio were used; the comparative computations with DDA showedthat spheroids can be exploited as a model of cirrus ice particles to study the interaction ofmillimetric polarized radiation. Spheroids have been used also by Mishchenko [24], Hill et al. [25]and by Lumme and Rahola [8].

In the present work, generalizing the areal distance presented in Stoyan and Stoyan [26], we usea volumetric distance: we consider that two shapes are closest to one another when their commonvolume is maximized. This metric has the advantage of being independent of the reference frame.Finding the best-"tting ellipsoid, according to this metric, is a multiparameter minimizationproblem with eight free parameters (three Euler angles, three coordinates of the center of mass ofthe ellipsoid, three semiaxes of the ellipsoid, one condition of equal volume), all calculated using thesimplex method [27]. The resulting orientation of the ellipsoid is not utilized in our followingcomputations since we average over orientation, but it could be necessary in the case of a singleorientation; in the same way, the center of mass was not utilized as single scattering properties aretranslationally invariant.

In the literature (e.g., Refs. [3,28]) it is common to introduce a class of equivalence for thescatterers: two scatterers are equivalent if their volumes are equal. It is necessary to test if twobodies with equal volume have equal or similar scattering patterns, leading us to examine whenshapes become relevant (this will be particularly e!ective in the comparison of the JSCAT, DDA,REA, and RVA computations). We already know that this will be true only in some regions of thex}m-plane because, for example, it is certainly false in the domain of geometric optics. Thus, webegin just from the opposite side of the Rayleigh regime, where we know that the absorption iscompletely equivalent to that of an equal-volume spherical scatterer.


Note that the current choice of equivalence among scatterers is supported also by the fact thatthe volume can, in some cases, be retrieved by an independent measurement (e.g., ice crystals forwhich the water content has been retrieved [29]).

Generally, we can "nd all the scattering parameters for any shape in the Rayleigh-ellipsoidapproximation. The problem is to understand when such an approximation can give reliableresults: we will start by testing the Rayleigh region hoping to obtain a good approximation,because, in this region, we have an analytical solution for ellipsoids. As a further step to bedeveloped, we could upgrade this model to resonance region using numerical solutions like¹-matrix codes (e.g., see Schneider and Peden [30], Laitinen and Lumme [31]).

3.3. Application to Gaussian particles

We demonstrate the method of best-"tting ellipsoids for Gaussian random particles. Thus, wederive the best-"tting ellipsoid for each realization of the Gaussian particle with di!erent para-meters and compare the scattering properties. In Fig. 3, we show some examples of Gaussianshapes along with their best-"t ellipsoids.

Since the Gaussian random particle is implicitly randomly oriented, we can speed up theconvergence of the Rayleigh-ellipsoid approximation by making use of orientationally averagedresults for each best-"t ellipsoid. We have averaged over hundreds of random realizations untilreaching error bars negligible as compared to the typical measurement errors. For cross sections,for example, we have obtained

SCM!"4

T"kT< ImA3+i/1

e1!e

m3e

m#3¸

i(e1!e

m)BU, (28)

SCM4#!

T"k4

2pT<23+i/1K

e1!e

m3e

m#3¸

i(e1!e

m) K

2

U, (29)

where S T stands for the ensemble average. So we have averages that are more complicated thanthose in Eqs. (19)}(20), connecting volume, refractive index, and the geometric factors ¸

i.

As a parameter of the goodness of the approximation and of the proximity of the two shapes, wehave calculated the ratio between the uncommon volume<X!<W and the volume of the Gaussianparticle by de"ning the `badnessa parameter

B"T<X!<W

< U. (30)

In the limit of BP0, we "nd exact results in the Rayleigh regime. Clearly, badness increases withp and when passing from ellipsoidal shapes to very sharp-edged ones (see Table 1). For example,particles P

2are better approximated because they have an almost ellipsoidal behavior. The same is

true for P1

particles and particles with !#"903 that are almost spherical. Therefore for these

particles, we will be very con"dent that we get realistic results. Note that for them the REA methodcan be used for testing other computational methods. However, !

#"103 and P

3particles are badly

approximated since they have lots of hills and valleys.


Fig. 3. The best-"t ellipsoids for the Gaussian particles in Fig. 1. Labels (a}f ) as in Fig. 1. In (a), (b) and (f ) cases, best-"tellipsoids are almost indistinguishable from the true shapes.


Table 1The badness and axial ratio with standard deviations as de"ned by Eqs. (30)}(31) for Classes-I and II Gaussian particles.We include the "rst and second moments of volume, computed exactly and from the sample particles used in thediscrete-dipole and Rayleigh-ellipsoid simulations

type p B% R < S<TDDA

S<TREA

<2 S<2TDDA

S<2TREA

103 0.1 11.2$0.7 1.09$0.04 4.316$0.16 4.328 4.307 18.65$1.37 18.76 18.580.2 21.7$1.4 1.20$0.09 4.712$0.36 4.741 4.687 22.33$3.39 22.62 22.090.3 31.5$2.1 1.33$0.12 5.425$0.64 5.474 5.392 29.85$7.02 30.44 29.48

303 0.1 6.4$1.1 1.21$0.08 4.316$0.49 4.352 4.311 18.85$4.40 19.19 18.830.2 12.5$2.2 1.45$0.21 4.712$1.2 4.794 4.733 23.35$12.4 24.21 23.720.3 18.2$3.7 1.72$0.36 5.425$2.0 5.567 5.417 33.19$26.2 34.81 33.39

903 0.1 1.0$0.3 1.09$0.04 4.316$1.0 4.404 4.304 19.72$9.66 20.67 19.570.2 2.1$0.7 1.166$0.066 4.712$2.5 4.934 4.682 27.81$33.7 31.24 27.370.3 3.7$1.7 1.23$0.1 5.425$4.4 5.858 5.369 48.53$96.1 58.48 47.66

P1

0.1 0.12$0.15 1.002$0.004 4.316$0.15 4.312 18.65$1.35 18.620.2 0.22$0.56 1.005$0.016 4.712$0.66 4.697 22.66$7.05 22.490.3 0.37$0.56 1.01$0.11 5.425$1.8 5.442 32.46$28.5 32.70

P2

0.1 1.0$0.6 1.42$0.15 4.316$0.12 4.307 4.315 18.64$1.04 18.56 18.630.2 3.5$2.1 1.915$0.33 44.712$0.48 4.674 4.718 22.49$5.52 22.10 22.550.3 6.5$3.4 2.39$0.50 5.425$1.3 5.334 5.431 31.37$19.3 30.02 31.47

P3

0.1 11.5$3.1 1.024$0.025 4.316$0.10 4.314 18.64$0.90 18.620.2 22.0$5.5 1.150$0.16 4.712$0.44 4.706 22.4$4.4 22.340.3 30.5$6.6 1.41$0.37 5.425$1.1 5.409 30.7$14.1 30.49

Another interesting quantity is the axial ratio R de"ned as the ensemble average of the ratio ofthe longest (a) and shortest (c) axes of the best-"tting ellipsoid,

R"TacU. (31)

This parameter gives a good idea of how far we are from the best-"tting sphere approximation.From Table 1 we can deduce that it is not so useful to compute the scattering characteristics for

!"903 and P1

particles in REA, since in this case REA and RVA are quite the same, while just theopposite is true for !"303 and P

2particles. For !"103 and P

3particles we have great badness

and low axial ratio, that is, the best-"tting ellipsoid is not so good and is nearly spherical.

4. Other methods for veri5cation

Gaussian particles' scattering problem cannot be solved analytically. Therefore, in order to testour model, we have to compare our results with other numerical methods. We brie#y summarizethese techniques.


4.1. Rayleigh-volume approximation

The RVA scattering characteristics are just those of a population of spheres with radiiri"J3<

i/4p, <

ibeing the volume of the ith realization, computed in the Rayleigh approximation.

For example, the cross sections are [3],

SC!"4

T"3kS<TImAe1!e

me1#2e

mB, (32)

SC4#!

T"32p

k4S<2TKe1!e

me1#2e

mK2, (33)

where S<T and S<2T are given in Table 1, while the scattering matrix is the same as in Eq. (21).

4.2. Discrete-dipole approximation

The discrete-dipole approximation (DDA) is a method in which the scatterer is replaced by anarray of polarizable points, the dipoles. DDA contains two approximations: "rst, the scatterer isrepresented by the dipole array and, second, approximations must be introduced to computethe dipole polarizability. In principle, DDA can be used to compute the scattering by anarbitrary target. In practice, the accuracy of the method is limited by the computer memory andthe computational capacity, limiting the number of dipoles used approximating the scatterer.This limits the usability of the method to size parameters x(10 and relatively low refrac-tive incides (Dm!1D43). On the other hand, the method works also with anisotropic targets.For more information about the discrete-dipole approximation, see for example, Draine andFlatau [17].

In the present study, the DDA method is applied only in the cases of low refractive index. Latticedispersion relation (Draine and Goodman [32] and references therein) and GPFA FFT algorithmby Temperton are applied in the simulations. The DDSCAT code by Draine and Flatau is modi"edfor Gaussian random spheres and ensemble averaging. This is accomplished by writing severalexternal applications to handle the shape generation, computation of the volume of the targets, andensemble averaging (in our case we have used 144 orientations). It is noteworthy that the originalDDSCAT model computes the volume of the scatterer from the given e!ective radius and thenumber of dipoles in order to conserve the volume in the volume discretization process. Thus, forGaussian random shapes with varying volume, the e!ective radius needed to be modi"ed for eachrealization.

4.3. Variational volume integral technique

In the variational volume integral equation technique, one divides the scatterer to smallelements, expands the electric "eld inside these elements using a low-order vector spherical waveexpansion, substitutes the expansion to the integral form of the Maxwell equations, and solves forthe unknown expansion coe$cients in the least-squares sense. The very careful treatment of thesingularity of the dyadic Green function, and e$cient numeric integration are essential parts of thetechnique.


Peltoniemi's [2] JSCAT code used here utilizes full second-order treatment of the singularity (incontrast to most DDA codes that account for only part of the second-order terms). Thus it worksalso for large refractive indices. Especially, we remark that JSCAT is apparently free of any knowninherent inexactness, i.e., it converges towards the correct result (at least in cases where it is known),given enough computing time and memory.

Here, we divide our Gaussian particle to about 90}130 cells, and select an expansion includingelectric and magnetic dipoles, i.e. six terms per element. For !

#"30 and 903 this is su$cient, but

!#"103 yields such a "ne surface structure that higher orders would be needed, but that would

also increase the computing time beyond our patience. We have averaged over 99 di!erentorientations.

4.4. Second-order perturbation approximation

Of utmost importance in the resonance region is the analytical, second-order perturbationapproximation [14] (PS2) for light scattering by statistically irregular particles with surfacedeformations much smaller than the mean radius of the particle and much smaller than thewavelength.

In order to make the perturbation approximation [14,33,34] applicable to Gaussian randomparticles, we must rephrase the shape as [15]

r(0,u),r6 [1#f (0,u)],

f (0,u)"expCs(0, u)!12b2D!1, (34)

so that

S f (0, u)T"0, S f (01,u

1) f (0

2, u

2)T"&

r(c). (35)

Note that the random variable f is lognormally distributed with zero mean and with the so-called threshold equal to !1. Legendre expansion of the radius covariance function is re-quired,

&r(c)"

=+l/0

DlPl(cos c),

Dl"Al#

12BP

1

~1

dmPl(m)&

r(m), m"cos c. (36)

For small surface deformations, the Dl-coe$cients come close to the C

l-coe$cients of the logradius

covariance function.Absorption and scattering cross sections can be computed from the extinction cross section and

the incoherent and coherent scattering cross sections,

C!"4

"C%95!C

4#!, C

4#!"C*/#0)

4#!#C#0)

4#!. (37)


With the help of the 3j symbols, Schi!er's J(m) and H(m)-functions (superscript denoting refractiveindex), and the zeroth-order Lorenz}Mie scattering coe$cients a(0)

l,1and b(0)

l,1, the cross sections are

C%95

"

2pk2

Re+nlL

(!1)lDL(2n#1)(2l#1)A

n l ¸

!1 1 0B2,

[J(M)1

(n, l,¸)#J(M)2

(n, l,¸)],

C*/#0)4#!

"

pk2

+nlL

DL(2n#1)(2l#1)A

n l ¸

!1 1 0B2

[DH(M)1

(n, l,¸)D2#DH(M)2

(n, l,¸)D2],

C#0)4#!

"

2pk2

Re+nlL

(!1)lDL(2n#1)(2l#1)A

n l ¸

!1 1 0B2

[a(0)Hl,1

J(M)1

(n, l,¸)#b(0)Hl,1

J(M)2

(n, l,¸)]. (38)

For a complete description of the second-order perturbation approximation, the reader is referredto the work by Schi!er [14].

5. Numerical methods

The REA program consists of two parts: "rst, we derive the best-"tting ellipsoid and, second,compute the orientationally averaged scattering characteristics for the ellipsoid. To obtain theensemble averages, the aforedescribed procedure is repeated for a given number of realizations.

We have de"ned two reference frames: the Gaussian particle is de"ned in the laboratory frameK@, while K is the natural frame of the ellipsoid. The transformation between the two frames is

Ax

y

zB"Ax#

y#

z#B#Eu(a

E,b

E, c

E)A

x@

y@

z@B, (39)

where the subscript c gives the center of mass of the Gaussian sample particle.To calculate analytically the expression of r@

%--(h@,/@) in K@, it is su$cient to intersect the ellipsoid

r%--

(h,/) (Eq. (14)) with the line

Ax

y

zB"Ax#

y#

z#B#jEu(a

E,b

E, c

E)A

sin h@cos/@

sin h@sin/@

cos h@ B,"nding j"r@

%--(h@,/@) as the positive root of the resulting second-degree equation. The common

volume of the two particles becomes

<W"13 P

1

~1P

2p

0

min[r@%--

3(h@,/@); r@3(h@,/@)] d cos h@d/@ (40)


and has been evaluated numerically using a 32-point Gaussian quadrature both in h and / direc-tions. Note that, in principle, it is straightforward to change the approximating shape to a cylinderor a spheroid, for which the problem has seven degrees of freedom.

The common volume <W in Eq. (40) has been maximized using the simplex method [27], takingparticular care of two constraints: the value of the semiaxes must be positive and the center of massof the ellipsoid must be inside the Gaussian particle. To make the convergence of the numericalmethod faster we have used the Karhunen}Loeve statistical ansatz [21] to construct the initialsimplex with a little change: instead of using surface points, we have taken N

P"104 randomly

distributed points inside the volume of the particle and we have computed the matrix

"ij"

1N

P!1

NP

+l/1

(ril!SriT)(rj

l!SrjT), SriT"

1N

P

NP

+l/1

ril, i, j"1,2,3,

where rilis the coordinate i of the point l. We can diagonalize this 3]3 symmetric matrix " with an

orthogonal transformation: the three eigenvectors give the directions of the semiaxes while fromthe three eigenvalues j

iwe obtain the semiaxes a

iby scaling them until the right volume is

produced. The center of mass of the initial ellipsoid is easily found by computing the center of massof the N

Ppoints. The initial simplex is produced by introducing small variations to the initial

ellipsoid. We set the tolerance of the simplex equal to 10~4 that guarantees convergence toa solution for every shape. Averaging the "nal simplex, we have obtained the three semiaxes, thethree Euler angles and the center of mass. Note that this part of the program depends only onparticle shape and can thus be run once for all refractive indices.

In the second part of the program we compute the cross sections and the scattering matricesanalytically, except for the computation of integrals in Eq. (16), where we used Gauss}Laguerreintegration with 100 integration points. Averaging over a consistent number of sample particles, we"nally found the scattering matrix and the cross sections for the Gaussian particles underconsideration.

For all scattering and absorption parameters, we computed the relative error de"ned asp/JN

4*.. On one hand, for a few thousand sample particles, the statistical errors of the average

cross sections are of the order of few percent. On the other hand, the ratios of two scattering matrixelements are less a!ected by the errors. This is due to a cancellation of errors caused by a positivecovariance between these elements. For example, fewer realizations are necessary to have goodestimates of the degree of polarization: with a few thousand particles, the errors are less than 0.1%.Another interesting aspect is that the error of p

4#!follows the one in the determination of S<2T

while the one of p!"4

follow the one in the determination of S<T (see Table 1). Therefore, theabsorption cross sections will be more accurately determined. To obtain the same accuracy forboth cross sections, we must increase N

4*.both when increasing p and changing the particle shape

(varying !#from 10 to 903 and passing from P

3to P

1): in fact, by so doing, there is an increase of the

ratios Dev(V)/S<T and Dev(V2)/S<2T, where &Dev' stands for standard deviation. For example, for!#"303 and p"0.3, we have to generate nearly 10 times more realizations than for !

#"303 and

p"0.1.The computational technique is very fast: the greatest part of the CPU time is consumed by the

simplex procedure that, for every realization in the most di$cult cases, can take up to 1 min on anINDIGO 2 SGI R4400 MIPS workstation at 250 MHz. Much of the consumption is, however, due


to the fact that we have made a very conservative choice of the simplex tolerance and the number ofquadrature points. The second part of the program is very fast (less than 0.2 s for each simulation).

The speed of the REA program allowed us to make computations with more than 1000 sampleshapes for each Gaussian particle. However, the considerable CPU time required by DDA andJSCAT has obliged us to take only 200 and 100 realizations of the Gaussian particles in DDA andJSCAT, respectively. Thus, to improve the comparison of the results, we took into account thedi!erent realizations of the Gaussian particles. In Table 1 we indicated the values of S<T and S<2Tof every sampling set used for the computations (see Table 1 also for the true value); correcting allcross sections by a factor S<2T/S<2T

4!.1-%for scattering and S<T/S<T

4!.1-%for absorption is

a good method, as the behavior of these quantities is given, in the "rst approximation, just byEq. (32) and by Eq. (33). The renormalized values are written in parentheses in the Tables 2}5.

6. Results and discussion

We simulated our two types of Gaussian particles with four complex refractive indices ofm"1.33#i0.01, 1.55#i0.01, 1.782#i0.004, and 3#i4, representing typical values both for as-trophysics and atmospheric physics (for example, the third one is refractive index of ice at 94 GHz).In all simulations we have taken the size parameter x"0.01, well inside the Rayleigh region for therefractive indices used.

6.1. Scattering and absorption cross sections

The absorption and scattering cross sections as computed with di!erent methods are sum-marized in Tables 2}5.

As regards REA, it is possible to see that, except the case of no absorption, the terms

h(¸1,¸

2,¸

3),

3+i/1K

e1!e

m3e

m#3¸

i(e1!e

m)K2,

g(¸1,¸

2,¸

3),ImA

3+i/1

e1!e

m3e

m#3¸

i(e1!e

m)B, (41)

that are present in Eqs. (28) and (29), satisfy the relation

h(¸1,¸

2,¸

3)

h(1/3, 1/3, 1/3)"

g(¸1,¸

2,¸

3)

g(1/3, 1/3, 1/3)(42)

and that, making use of the so-called Lagrange multipliers method, they reach their minima forperfect spheres (¸

i"1

3, i"1, 2, 3). In particular, for our refractive indices this is a global minimum

but, looking at Fig. 2 for functions (41), the minimum is very #at for small refractive indices, andbecomes more marked with high refractive indices. These considerations show that the REA crosssections are greater than the RVA analogous ones, con"rming the well-known fact that thespherical shape minimizes cross sections. In all the cases analyzed, the normalized results for cross


Table 2Scattering cross sections for Class-I particles using di!erent computational techniques divided by the values in theRayleigh-volume approximation. The values in parentheses are renormalized by the ratio S<2T/S<2T

5%#)/*26%

m !#

p p4#!

[REA] p4#!

[DDA] p4#!

[JSCAT]

1.33#i0.01 103 0.1 0.9975$0.0031 (1.0012) 1.0194$0.0057 (1.0063)0.2 0.9903$0.0052 (1.0011) 1.0398$0.0119 (1.0265)0.3 0.9904$0.0062 (1.0028) 1.0567$0.0191 (1.0362)

303 0.1 0.9999$0.0074 (1.0010) 1.0252$0.0168 (1.0070)0.2 1.0208$0.0188 (1.0049) 1.0509$0.0351 (1.0374)0.3 1.0158$0.0191 (1.0097) 1.0698$0.0546 (1.0200)

903 0.1 0.9932$0.0111 (1.0003) 1.0531$0.0399 (1.0047)0.2 0.9852$0.0271 (1.0010) 1.1288$0.0977 (1.0049)0.3 0.9839$0.0560 (1.0019) 1.2110$0.1807 (1.0051)

1.55#i0.01 103 0.1 0.9980$0.0031 (1.0018) 1.0284$0.0087 (1.0429)0.2 0.9919$0.0052 (1.0026) 1.0224$0.0177 (1.0589)0.3 0.9948$0.0062 (1.0073) 1.0100$0.0269 (1.0760)

303 0.1 1.0018$0.0074 (1.0029) 1.0420$0.0262 (1.0403)0.2 1.0281$0.0190 (1.0121) 1.0743$0.0580 (1.0549)0.3 1.0306$0.0195 (1.0244) 1.1233$0.1032 (1.0713)

903 0.1 0.9936$0.0111 (1.0007)0.2 0.9864$0.0271 (1.0023)0.3 0.9862$0.0562 (1.0042)

1.782#i0.004 103 0.1 0.9985$0.0031 (1.0023) 1.0634$0.0059 (1.0572) 1.0335$0.0088 (1.0487)0.2 0.9941$0.0052 (1.0049) 1.1320$0.0130 (1.1175) 1.0510$0.0183 (1.0855)0.3 1.0003$0.0062 (1.0128) 1.1888$0.021 (1.1658) 1.0375$0.0279 (1.1053)

303 0.1 1.0041$0.0074 (1.0052) 1.0486$0.0172 (1.0300) 1.0453$0.0264 (1.0436)0.2 1.0374$0.0193 (1.0212) 1.0966$0.0367 (1.0576) 1.0884$0.0591 (1.0687)0.3 1.0500$0.0200 (1.0437) 1.1416$0.0583 (1.0885) 1.1507$0.1067 (1.0975)

903 0.1 0.9940$0.0111 (1.0011) 1.061$0.0404 (1.0122) 1.0157$0.0545 (0.9966)0.2 0.9878$0.0272 (1.0037) 1.1455$0.0990 (1.0197) 1.0326$0.1266 (1.0382)0.3 0.9891$0.0564 (1.0071) 1.2305$0.1833 (1.0213) 1.0744$0.2415 (1.0455)

3#i4 103 0.1 1.0045$0.0031 (1.0083) 1.0659$0.0994 (1.0816)0.2 1.0194$0.0055 (1.0305) 1.0096$0.1876 (1.0500)0.3 1.0704$0.0072 (1.0838) 0.8871$0.2496 (0.9615)

303 0.1 1.0316$0.0078 (1.0327) 1.0832$0.0288 (1.0814)0.2 1.1627$0.0236 (1.1446) 1.2696$0.0738 (1.2466)0.3 1.3678$0.0341 (1.3596) 1.5148$0.1528 (1.4447)

903 0.1 0.9988$0.0112 (1.0059) 1.0052$0.0541 (1.0244)0.2 1.0045$0.0276 (1.0206) 1.0361$0.1286 (1.0417)0.3 1.0247$0.0595 (1.0434) 1.1030$0.2544 (1.0733)

sections (the values in parentheses) are always larger than in RVA. Besides, the ratio of crosssections in REA to cross sections in RVA increases as the elongation or the refractive indexincrease. This is as we have expected, and consistent with Fig. 2. Note also that REA normalizedvalues of scattering and absorption cross sections are very similar; this is a consequence of relation


Table 3As in Table 2 for absorption cross sections, except that the values in parentheses are now renormalized by the ratioS<T/S<T

5%#)/*26%

m !#

p p!"4

[REA] p!"4

[DDA] p!"#

[JSCAT]

1.33#i0.01 10" 0.1 0.9989$0.0015 (1.0010) 1.0226$0.0724 (1.0198)0.2 0.9957$0.0026 (1.0011) 1.0454$0.0742 (1.0390)0.3 0.9969$0.0031 (1.0030) 1.0629$0.0758 (1.0534)

303 0.1 1.0020$0.0036 (1.0032) 1.0187$0.0725 (1.0103)0.2 1.0093$0.0087 (1.0048) 0.9865$0.0715 (1.0182)0.3 1.0081$0.0084 (1.0096) 1.0537$0.0789 (1.0268)

903 0.1 0.9976$0.0053 (1.0004) 1.0273$0.0750 (1.0068)0.2 0.9944$0.0111 (1.0008) 1.0545$0.0844 (1.0068)0.3 0.9914$0.0179 (1.0017) 1.0879$0.1003 (1.0075)

1.55#i0.01 103 0.1 0.9993$0.0015 (1.0014) 1.0074$0.0043 (1.0147)0.2 0.9974$0.0026 (1.0027) 1.0238$0.0091 (1.0427)0.3 1.0012$0.0031 (1.0073) 1.0350$0.0141 (1.0700)

303 0.1 1.0020$0.0036 (1.0032) 1.0056$0.0126 (1.0064)0.2 1.0163$0.0088 (1.0118) 1.0273$0.0265 (1.0240)0.3 1.0222$0.0086 (1.0237) 1.0535$0.0430 (1.0429)

903 0.1 0.9980$0.0084 (1.0008)0.2 0.9956$0.0111 (1.0020)0.3 0.9936$0.0180 (1.0040)

1.782#i0.004 103 0.1 0.9999$0.0015 (1.0020) 1.0879$0.0769 (1.0849) 1.0191$0.0045 (1.0265)0.2 0.9995$0.0026 (1.0048) 1.1820$0.0839 (1.1748) 1.0702$0.0097 (1.0886)0.3 1.0067$0.0031 (1.0129) 1.2578$0.0897 (1.2465) 1.0867$0.0151 (1.1235)

303 0.1 1.0043$0.0037 (1.0055) 1.0517$0.0748 (1.0430) 1.0104$0.0128 (1.0112)0.2 1.0252$0.0089 (1.0207) 1.0973$0.0796 (1.0785) 1.0456$0.0273 (1.0422)0.3 1.0405$0.0088 (1.0420) 1.1471$0.0859 (1.1178) 1.0867$0.0449 (1.0758)

903 0.1 0.9984$0.0053 (1.0012) 1.0492$0.0766 (1.0282) 0.9863$0.0253 (0.9990)0.2 0.9970$0.0111 (1.0034) 1.0779$0.0862 (1.0292) 0.9864$0.0529 (1.0022)0.3 0.9964$0.0180 (1.0068) 1.1136$0.1025 (1.0313) 0.9958$0.0861 (1.0071)

3#i4 103 0.1 1.0058$0.0016 (1.0079) 1.1937$0.0080 (1.2027)0.2 1.0249$0.0029 (1.0304) 1.2037$0.0146 (1.2287)0.3 1.0764$0.0040 (1.0830) 1.0998$0.0188 (1.1469)

303 0.1 1.0315$0.0039 (1.0327) 1.0851$0.0155 (1.0859)0.2 1.1434$0.0116 (1.1383) 1.3418$0.0401 (1.3375)0.3 1.3324$0.0159 (1.3344) 1.6395$0.0754 (1.6230)

903 0.1 1.0032$0.0053 (1.0060) 0.9806$0.0252 (0.9932)0.2 1.0139$0.0113 (1.0204) 0.9951$0.0538 (1.0110)0.3 1.0306$0.0188 (1.0413) 1.0263$0.0904 (1.0379)

(42). They are not perfectly equal because of the presence of ensemble average in Eqs. (28), (29).Only for the !

#"303 and P

2particles, there are relevant di!erences between REA and RVA: for

m"3#4i and p"0.3 up to &35% and &200%, respectively, both for scattering and absorp-tion. In all other simulations, the di!erences are considerable only for m"3#4i and greatelongation p"0.2, 0.3 but, however, always less than 10%.


Table 4As in Tables 2}3 for class-II particles P

1and P

3

m p p4#!

[REA] p4#!

[PS2] p!"4

[REA] p!"4

[PS2]

1.33#i0.01 P1

0.1 0.9985$0.0025 (1.0001) 0.9971 0.9993$0.0012 (1.0002) 0.99960.2 0.9923$0.0107 (0.9998) 0.9593 0.9970$0.0049 (1.0002) 0.99430.3 1.0075$0.0196 (1.0001) 0.8301 1.0033$0.0073 (1.0002) 0.9749

P3

0.1 0.9994$0.0010 (1.0005) 1.0036 0.9998$0.0005 (1.0003) 1.00850.2 0.9986$0.0044 (1.0013) 0.9886 0.9999$0.0021 (1.0012) 1.02620.3 1.0025$0.0106 (1.0094) 0.9043 1.0052$0.0048 (1.0082) 1.0359

1.55#i0.01 P1

0.1 0.9985$0.0025 (1.0001) 0.9971 0.9993$0.0012 (1.0002) 0.99970.2 0.9924$0.0107 (0.9999) 0.9593 0.9971$0.0049 (1.0003) 0.99440.3 1.0076$0.0197 (1.0002) 0.8301 1.0034$0.0073 (1.0003) 0.9749

P3

0.1 0.9995$0.0011 (1.0006) 1.0116 0.9999$0.0005 (1.0004) 1.02060.2 1.0005$0.0045 (1.0032) 1.015 1.0016$0.0022 (1.0029) 1.07000.3 1.0166$0.0111 (1.0236) 0.9467 1.0171$0.0051 (1.0201) 1.1195

1.782#i0.004 P1

0.1 0.9986$0.0026 (1.0002) 0.9971 0.9994$0.0012 (1.0003) 0.99970.2 0.9926$0.0107 (1.0001) 0.9593 0.9973$0.0049 (1.0005) 0.99450.3 1.0077$0.0197 (1.0003) 0.8301 1.0034$0.0073 (1.0003) 0.9749

P3

0.1 0.9996$0.0011 (1.0007) 1.0214 1.0000$0.0005 (1.0005) 1.03530.2 1.0028$0.0046 (1.0055) 1.0471 1.0038$0.0023 (1.0051) 1.12270.3 1.0349$0.0117 (1.0420) 0.9984 1.0324$0.0056 (1.0355) 1.2206

3#i4 P1

0.1 0.9987$0.0026 (1.0003) 0.9973 0.9995$0.0013 (1.0004) 1.0050.2 0.9945$0.0108 (1.0020) 0.9594 0.9990$0.0050 (1.0021) 0.99520.3 1.0088$0.0198 (1.0014) 0.8302 1.0042$0.0073 (1.0011) 0.9757

P3

0.1 1.0006$0.0011 (1.0017) 1.1228 1.0009$0.0006 (1.0014) 1.18390.2 1.0322$0.0058 (1.0350) 1.3804 1.0299$0.0033 (1.0312) 1.65850.3 1.3343$0.0266 (1.3435) 1.5349 1.2722$0.0149 (1.2760) 2.2455

DDA simulations agree with the REA ones within &2% for the best approximated shapes(!

#"903 and P

2) but there are considerable di!erences for !

#"103 (up to &15% for scattering

and &23% for absorption in the m"3#4i, p"0.3 simulation) and for !#"303 (analogously up

to &4 and &7%, respectively). The same is true for JSCAT: for !#"903 there are di!erences

within &3%, while for !#"103 we have di!erences up to 20 and 10% for absorption and

scattering, respectively. Though the DDA and JSCAT methods are a!ected by di!erent potentialsources of error, nevertheless these data give us the feeling that REA, although improving RVA,underestimates the cross sections; this underestimate increases with increasing badness, as expectedfrom geometrical considerations.

Comparisons with the second-order perturbation approximation are relevant only for p"0.1.For example, looking at the P

1particle (in this case we trust in REA since, in every case, B41%)

there are discrepancies up to 20%. On the other hand, for p"0.1, the results are perfectlycompatible. Therefore, we have also devised a preliminary test of the perturbation approximation,concluding that, for the Gaussian particles, it gives very accurate results for p"0.1 while departing


Table 5As in Table 4 for class-II P

2particles

m p p4#!

[REA] p4#!

[PS2] p4#!

[DDA]

1.33#i0.01 0.1 1.0036$0.0018 (1.0041) 1.0019 1.0036$0.0040 (1.0098)0.2 1.0159$0.0026 (1.0132) 0.9805 0.9978$0.0163 (1.0154)0.3 1.0316$0.0124 (1.0283) 0.8804 0.9791$0.0370 (1.0231)

1.55#i0.01 0.1 1.0097$0.0020 (1.0102) 1.00790.2 1.0355$0.0027 (1.0327) 1.00040.3 1.0651$0.0126 (1.0617) 0.9120

1.782#i0.004 0.1 1.0173$0.0021 (1.0178) 1.0155 1.0288$0.0046 (1.0327)0.2 1.0637$0.0029 (1.0609) 1.0254 1.0497$0.0185 (1.0682)0.3 1.1161$0.0137 (1.1126) 0.9517 1.0582$0.0422 (1.1058)

3#i4 0.1 1.1156$0.0043 (1.1162) 1.10370.2 1.5198$0.0079 (1.5158) 1.31440.3 2.3042$0.0524 (2.2969) 1.4101

p!"4

[REA] p!"4

[PS2] p!"4

[DDA]

1.33#i0.01 0.1 1.0039$0.0009 (1.0041) 1.0040 1.0077$0.0761 (1.0098)0.2 1.0133$0.0012 (1.0120) 1.010 1.0093$0.072 (1.0173)0.3 1.0255$0.0049 (1.0244) 1.0064 1.0077$0.0733 (1.0249)

1.55#i0.01 0.1 1.0098$0.0011 (1.0100) 1.01030.2 1.0332$0.0013 (1.0319) 1.03310.3 1.0595$0.0053 (1.0583) 1.0500

1.782#i0.004 0.1 1.0173$0.0012 (1.0175) 1.0182 1.0402$0.0735 (1.0424)0.2 1.0558$0.0015 (1.0545) 1.0616 1.0680$0.0761 (1.0765)0.3 1.1056$0.0059 (1.1044) 1.1045 1.0941$0.0731 (1.1128)

3#i4 0.1 1.114$0.0032 (1.1143) 1.10960.2 1.4803$0.0052 (1.4784) 1.39160.3 2.1065$0.0256 (2.1042) 1.7383

more and more from the correct solutions with increasing p. The tabulated values give an idea howlarge these departures are (also Fig. 4).

6.2. Scattering matrix elements

As to the scattering matrix elements, with all computational techniques, we con"rm theneutralization of the scattering matrix in the Rayleigh-volume approximation (Eq. (21)). Wecharacterize the polarization by !S

12(903)/S

11(903), S

22(03)/S

11(03), and S

44(03)/S

11(03), noting

the facts that S33

(03)"S22

(03) and S22

(903)"!S12

(903). The relevant data are reported inTables 6 and 7.


Fig. 4. Scattering by P2

particles with refractive index m"3#i4 and standard deviation p"0.3: (a) the degree oflinear polarization for incident unpolarized light !S

12/S

11and the ratio S

22/S

11in the Rayleigh-ellipsoid approxima-

tion (lower and upper solid lines, respectively) and in the second-order perturbation approximation (lower and upperdotted lines). (b) the scattering matrix element ratios S

33/S

11and S

44/S

11in the Rayleigh-ellipsoid approximation (solid

and dash-dotted lines, respectively) and in the second-order perturbation approximation (bold dotted and dotted lines,respectively).


Table 6Polarization parameters for the class-I !

#"303 particles. Statistical errors are smaller than 10~4 for p"0.1, than

2]10~3 for p"0.2 and than 6]10~3 for p"0.3 and usually increase with increasing m

!S12

/S11

(903) S22

/S11

(03) S44

/S11

(03)

m p [REA] [DDA] [JSCAT] [REA] [DDA] [JSCAT] [REA] [DDA] [JSCAT]

1.33#i0.01 0.1 0.9995 0.9996 0.9998 0.9998 0.9995 0.99960.2 0.9980 0.9987 0.9990 0.9992 0.9980 0.99870.3 0.9961 0.9976 0.9980 0.9987 0.9961 0.9975

1.55#i0.01 0.1 0.9988 0.9988 0.9994 0.9994 0.9989 0.99890.2 0.9952 0.9954 0.9976 0.9986 0.9952 0.99610.3 0.9903 0.9920 0.9951 0.9964 0.9903 0.9928

1.782#i0.004 0.1 0.9979 0.9984 0.9978 0.9989 0.9989 0.9990 0.9979 0.9985 0.99800.2 0.9916 0.9943 0.9926 0.9958 0.9973 0.9966 0.9916 0.9942 0.99320.3 0.9829 0.9894 0.9860 0.9914 0.9945 0.9936 0.9828 0.9887 0.9873

3#i4 0.1 0.9869 0.9872 0.9934 0.9939 0.9868 0.98800.2 0.9457 0.9550 0.9721 0.9780 0.9442 0.95610.3 0.8782 0.9160 0.9352 0.9590 0.9442 0.9561

Table 7Polarization parameters for the class-II P

2particles. Statistical errors are smaller than 10~4 for p"0.1, than 5]10~3 for

p"0.2 and than 5]10~3 for p"0.3

!S12

/S11

(90") S22

/S11

(0") S44

/S11

(0")

m p [REA] [PS2] [DDA] [REA] [PS2] [DDA] [REA] [PS2] [DDA]

1.33#i0.01 0.1 0.9984 0.9981 0.9987 0.9992 0.9989 0.9993 0.9984 0.9983 0.99880.2 0.9947 0.9928 0.9962 0.9974 0.9956 0.9979 0.9947 0.9943 0.99580.3 0.9904 0.9849 0.9941 0.9952 0.9899 0.9967 0.9904 0.9900 0.9933

1.55#i0.01 0.1 0.9960 0.9955 0.9980 0.9975 0.9960 0.99590.2 0.9870 0.9844 0.9935 0.9910 0.9869 0.98650.3 0.9763 0.9706 0.9880 0.9821 0.9760 0.9764

1.782#i0.004 0.1 0.9930 0.9923 0.9946 0.9965 0.9958 0.9969 0.9930 0.9929 0.99430.2 0.9772 0.9743 0.9838 0.9885 0.9855 0.9907 0.9769 0.9770 0.98180.3 0.9581 0.9541 0.9743 0.9786 0.9729 0.9849 0.9572 0.9605 0.9698

3#i4 0.1 0.9560 0.958 0.9775 0.9769 0.955 0.96050.2 0.8494 0.8855 0.9186 0.9346 0.8371 0.88930.3 0.7188 0.8314 0.8364 0.9004 0.6728 0.8341


Fig. 5. Scattering by !#"303 particles with refractive index m"3#i4 and standard deviation p"0.2: (a) the degree

of linear polarization !S12

/S11

; (b) the ratios S22

/S11

(solid line), S33

/S11

(dotted line), S44

/S11

(dash}dotted line).

The polarization parameters depart more and more from the Rayleigh-volume approximationwith increasing standard deviation (and thus average elongation) and refractive index. The REAand RVA parameters for the !

#"10 and 903, P

1, and P

3particles are nearly indistinguishable. For

the P1particles, the di!erences are always less than 0.1%, while the !

#"103 and !

#"903 particles


present di!erences 43 and 42%, respectively. For example, for !#"10 and 903 with m"3#i4

and p"0.3, !S12

(903)/S11

(903)"0.967 and 0.983, respectively. For the P3particles, the REA and

RVA are within 2% from each other, except for the m"3#i4 and p"0.3 case, for which!S

12(903)/S

11(903)"0.885, S

22(03)/S

11(03)"0.939, and S

44(03)/S

11(03)"0.878.

For the !#"303 and P

2particles, we "nd signi"cant e!ects. As an example, in Fig. 4), we plot the

polarization and the diagonal terms S22

, S33

, S44

for the P2

particles with refractive indexm"3#i4 and standard deviation p"0.3 (Fig. 4) computed in REA and in PS2. Note the largedi!erence between these two models just because here we have a large p. As another example, weplot the same parameters for the !

#"303 particle but with m"3#i4 and p"0.2 in REA (Fig. 5).

Generalizing from the comparison between di!erent computational techniques, we can say thatREA underestimates polarization parameters.

Thus, di!erences from RVA are expected for high refractive indices (DmD52) and for particleswith pronounced elongation (R51.5). In the latter case, the ellipsoidal approximation gives resultsquite di!erent from RVA. Note, however, that the high refractive index and considerable elonga-tion alone are conditions necessary but not su$cient to give behavior di!erent from spheres. Ina certain sense, the di!erence between REA and RVA proves that the shapes become importantwith high elongation and refractive index. For shapes that are more rough, REA fails to improveRVA, because the best-"t ellipsoids are very spherical.

7. Conclusion

We have provided a fairly thorough treatment of light scattering by Gaussian particles in theRayleigh-ellipsoid approximation, upgrading the well-known Rayleigh-volume approximation.After de"ning the best-"t ellipsoid and revisiting our implemented code, we have made numerouscomparisons with other codes. Only for the most elongated and least rough Gaussian shapes do we"nd results di!ering considerably from those of RVA and, in these cases, the results are partially inagreement with the other computational techniques even if, in general, REA seems to underesti-mate both cross sections and polarization parameters.

A natural extension of REA could be the corresponding ellipsoid approximation in the reson-ance region; we are currently working to upgrade our method from the Rayleigh to the resonanceregion using the general ¹-matrix method [30,31]. Therefore, thanks to its speed and easiness, webelieve that our equal-volume-ellipsoid method can be a useful tool for current scattering studies inthe Rayleigh region, and for future studies in the resonance region.

Acknowledgements

We are grateful to Prof. F. Prodi for initiating and supporting the present collaboration betweenthe Cloud & Precipitation Group of FISBAT in Bologna, the Stellar-Planetary Astronomy Groupat the Observatory, University of Helsinki, the Department of Meteorology, University of Helsinki,and the Finnish Geodetic Institute. Dr. L. Lamberg pointed out the Karhunen}Loeve algorithmfor the computation of the initial approximation of the best-"t ellipsoid.


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